Rationalize the Denominator

"Rationalizing the denominator" is when we move a root (like a square root or cube root) from the bottom of a fraction to the top.

Oh No! An Irrational Denominator!

The bottom of a fraction is called the denominator.
Numbers like 2 and 3 are rational.
But many roots, such as √2 and √3, are irrational.

Example: has an Irrational Denominator

To be in "simplest form" the denominator should not be irrational!

Fixing it (by making the denominator rational)
is called "Rationalizing the Denominator"

Note: there is nothing wrong with an irrational denominator, it still works. But it is not "simplest form" and so can cost you marks. And removing them may help you solve an equation, so you should learn how.

So ... how do we do it?

1. Multiply Both Top and Bottom by a Root

Sometimes we can just multiply both top and bottom by a root:

Example: has an Irrational Denominator. Let's fix it.

Multiply top and bottom by the square root of 2, because: √2 × √2 = 2: